Question

The statement having truth value 'T' from the following is

• A. \(sin(x)\) is an even function.
• B. Every square matrix is non-singular.
• C. The product of complex number and its conjugate is purely imaginary.
• D. A square of any real number is non-negative.

Hint:

A statement is true only if it holds for every case; test each one with a quick example.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
A statement has truth value T if it is always true. A single counter-example makes it false.

Step 2: Check statement (A)
\(\sin(-x)=-\sin x\), so \(\sin x\) is an odd function. For example, \(\sin(-30^\circ)=-\tfrac12\) but \(\sin30^\circ=\tfrac12\). So "even" is false.

Step 3: Check statement (B)
A square matrix is non-singular only when its determinant is not zero. The matrix \(\begin{bmatrix}1&1\\1&1\end{bmatrix}\) has determinant 0, so it is singular. So (B) is false.

Step 4: Check statement (C)
For \(z=a+ib\), \(z\bar z=a^2+b^2\), which is a real number. For \(z=1\) it is 1, which is not purely imaginary. So (C) is false.

Step 5: Check statement (D)
For any real \(x\), \(x^2\ge0\). So (D) is true.

Final Answer:
Only the statement that the square of any real number is non-negative is always true, option (D). \[ \boxed{\text{(D)}} \]